Thaum twg yog ib pab pawg neeg abelian cyclic?

2024 Tus sau: Fiona Howard | [email protected]. Kawg hloov kho: 2024-01-10 06:38

Kev tsis raug cai, ib pawg yog cyclic yog tias nws yog tsim los ntawm ib lub ntsiab. Nws yog abelian yog tias kev sib tw sib tw. Ib pawg yog cyclic yog tias nws tuaj yeem tsim los ntawm ib lub caij.

Puas yog pawg abelian cyclic?

Txhua pawg neeg caij tsheb kauj vab yog Abelian, tab sis ib pab pawg Abelian tsis tas yuav mus ncig. Txhua pawg ntawm pawg Abelian yog ib txwm muaj. Hauv ib pab pawg Abelian, txhua lub ntsiab lus yog nyob rau hauv chav kawm conjugacy los ntawm nws tus kheej, thiab lub cim lub rooj koom nrog lub hwj chim ntawm ib lub ntsiab lus hu ua pab pawg neeg tsim khoom.

Koj ua pov thawj li cas pab pawg neeg abelian yog cyclic?

pov thawj

1. Cia G ua ib pab pawg uas muaj lub tshuab hluav taws xob g∈G. Namely, peb muaj G=⟨g⟩ (txhua yam hauv G yog qee lub zog ntawm g.)
2. Cia a thiab b ua arbitrary ntsiab hauv G. Ces muaj n, m∈Z xws li a=gn thiab b=gm.
3. Yog li peb tau txais ab=ba rau arbitrary a, b∈G. Yog li G yog ib pab pawg neeg abelian.

Yuav ua li cas koj thiaj paub tias ib pab pawg yog cyclic?

4 Teb. Ib pawg finite yog cyclic yog, thiab tsuas yog, nws muaj ib pawg ntawm txhua pawg ntawm nws qhov kev txiav txim. Yog li yog tias koj pom ob pawg ntawm tib pawg, ces pawg tsis yog cyclic, thiab qhov ntawd tuaj yeem pab qee zaum.

Dab tsi yog pawg cyclic piav nrog piv txwv?

Piv txwv li, (Z/6Z)^×={1, 5} , thiab txij li 6 yog ob zaug qhov khib thawj qhov no yog ib pab pawg cyclic. … Thaum (Z/nZ)^× yog cyclic, nws cov generators hu ua primitive roots modulo n. Rau tus lej tseem ceeb p, pawg (Z/pZ)^{× yog ib txwm cyclic, suav nrog cov ntsiab lus tsis yog xoom ntawm qhov kawg ntawm kev txiav txim p.}